Can you turn a pose into the actual geometry of the mesh?
Yup.

Pose your armature.  Select your mesh.  Copy the armature modifier, then apply the first (topmost) copy.  Select your armature in pose mode and ctrl-a -> apply pose as resting pose.  Done.

There can be some issues if you have some constraints built around assumptions about the base pose.  It's not possible (to my knowledge) to apply a resting pose only to a single bone, so make sure your armature is in the pose you want before doing it.